EXPECTATION FORMULAS
7
27. CHI-SQUARE DISTRIBUTION
Begin with standard normal Z and let Z1 , Z2 , Z3 , ., Zd be an independent random sample
of size d. Dene Wd and 2 by
d
2
d
d
X
2
= fW 2 , Wd =
2
Zk
THE EXPECTATION PRIMER
EXPECTATION COVARIANCE AND PROBABILITY V3.2
7
that one way to arrive at a guessing procedure for expectation is to try to guess the long run
average, in the case of random varia
SETS & FUNCTIONS
3
3.6. SET INTERSECTION:. If A and B are sets, then their Intersection, denoted
A \ B, is the set
A \ B = cfw_x 2 A : x 2 B = cfw_x : x 2 A & x 2 B = cfw_x 2 B : x 2 A,
and thus con
SETS & FUNCTIONS
7
5.7. AXIOM OF CHOICE:. If C is a non-empty collection of non-empty sets, then
Y
C 6= ;.
5.8. DISTRIBUTIVE LAWS:. For any sets A, B, C,
A \ ( B [ C ) = (A \ B ) [ ( A \ C ) ,
A [ ( B
SETS & FUNCTIONS
9
5.20. MUTUALLY INVERSE FUNCTIONS:. We say that f and g are Mutually
Inverse Functions provided that g f is the identity on the domain of f and f g is
the identity on the domain of g
4
M. J. DUPRE
This will mean that in case of a continuous unknown, we can approximate to whatever
degree of accuracy we might require with unknowns which are discrete (notice again that
Ln (X ),Rn (X
THE EXPECTATION PRIMER
EXPECTATION COVARIANCE PROBABILITY V3.2
MAURICE J. DUPRE
1. INTRODUCTION
These notes are designed to complement an elementary course in probability and/or statistics. The aim is
SETS & FUNCTIONS
MAURICE J. DUPRE
1. SETS
1.1. SET:. an undened term. A set S can also be called a Collection.
1.2. SET MEMBERSHIP:. The statement that x is a member of set S is denoted x 2 S.
We also
EXPECTATION FORMULAS
11
Suppose now we have X1 , X2 , X3 , ., XnX and Y1 , Y2 , Y 3, ., YnY are independent random
samples for each random variable, so XnX or simply X is the sample mean random variab
EXPECTATION FORMULAS
Maurice J. Dupr
e
Department of Mathematics
New Orleans, LA 70118
email: [email protected]
November 2010
In the formulas given below, U, W, X, Y are any unknowns, and A, B, C, D,
EXPECTATION FORMULAS
3
11. DEFINITION OF COVARIANCE
X ] [ Y
Cov (X, Y ) = E ([X
Y ])
12. DEFINITION OF VARIANCE
V ar(X ) = Cov (X, X )
13. DEFINITION OF STANDARD DEVIATION
p
SD(X ) = X = V ar(X )
14.
EXPECTATION FORMULAS
23. DIRAC DELTA FUNCTION
Not really a function but it is denoted
with compact support
h(0) =
with the property that for any smooth function h
Z
1
h(x) (x)dx
1
c ( x)
h( c ) =
5
=
EXPECTATION FORMULAS
9
30. SAMPLING TO ESTIMATE THE MEAN
If X is a random variable with known standard deviation X but with unknown mean, not a
common circumstance, then in case Xn is normal, a conden
MODELO DE VALUACION Y RIESGO
INSTRUCCIONES
Con la informacin de la Compaa EL CAFETAL, que se proporciona realice lo siguiente:
a) El Flujo de Efectivo del proyecto para 10 aos
b) El Estado de Resultad